Angle sums of random simplices in dimensions $3$ and $4$
Abstract
Consider a random $d$dimensional simplex whose vertices are $d+1$ random points sampled independently and uniformly from the unit sphere in $\mathbb R^d$. We show that the expected sum of solid angles at the vertices of this random simplex equals $\frac 18$ if $d=3$ and $\frac{539}{288\pi^2}\frac 16$ if $d=4$. The angles are measured as proportions of the full solid angle which is normalized to be $1$. Similar formulae are obtained if the vertices of the simplex are uniformly distributed in the unit ball. These results are special cases of general formulae for the expected anglesums of random beta simplices in dimensions $3$ and $4$.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.01533
 Bibcode:
 2019arXiv190501533K
 Keywords:

 Mathematics  Probability;
 Mathematics  Metric Geometry;
 Primary: 52A22;
 60D05;
 Secondary: 52B11
 EPrint:
 9 pages